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Theorem cbvcllem 36934
 Description: Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
cbvcllem.y (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvcllem {𝑥 ∣ (𝑋𝑥𝜑)} = {𝑦 ∣ (𝑋𝑦𝜓)}
Distinct variable groups:   𝑥,𝑦,𝑋   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvcllem
StepHypRef Expression
1 cbvcllem.y . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21cleq2lem 36933 . 2 (𝑥 = 𝑦 → ((𝑋𝑥𝜑) ↔ (𝑋𝑦𝜓)))
32cbvabv 2734 1 {𝑥 ∣ (𝑋𝑥𝜑)} = {𝑦 ∣ (𝑋𝑦𝜓)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  {cab 2596   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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